Continuum modeling of crowd turbulence

With the growth in world population, the density of crowds in public places has been increasing steadily, leading to a higher incidence of crowd disasters at high densities. Recent research suggests that emergent chaotic behavior at high densities—known collectively as crowd turbulence—is to blame. Thus, a deeper understanding of crowd turbulence is needed to facilitate efforts to prevent and plan for chaotic conditions in high-density crowds. However, it has been noted that existing algorithms modeling collision avoidance cannot faithfully simulate crowd turbulence. We hypothesize that simulation of crowd turbulence requires modeling of both collision avoidance and frictional forces arising from pedestrian interactions. Accordingly, we propose a model for turbulent crowd simulation, which incorporates a model for interpersonal stress and acceleration constraints similar to real-world pedestrians. Our simulated results demonstrate a close correspondence with observed metrics for crowd turbulence as measured in known crowd disasters.

Figure 1

Continuum simulation algorithm for crowd turbulence.

Figure 2

Scene setup for Hajj simulations. The scene consists of two merging sets of agents emerging from the inlets shown in the blue checkerboard pattern, following the path highlighted by dotted arrows and exiting via the yellow wavy textured outlet on the right. Corridor dimensions are shown by solid red arrows.

Figure 3

(a) Plot of average speed vs density. Note that even though average speeds may be low at high densities, high variance implies that agent speeds may be significantly higher than the mean. (b) Plot of crowd “pressure”$P\left(\mathrm{x}\right)=\rho \left(\mathrm{x}\right){Var}_{\mathrm{x}}\left(\mathrm{v}\right)$ as defined by [1]. We observe values greater than $0.02$ at local densities of 7 people per ${\mathrm{m}}^2$ and higher, which are indicative of crowd turbulence.

Figure 4

Representative trajectories taken by agents in the Hajj simulation to travel a distance of 8 m in laminar (red, right), stop-and-go (black, middle), and turbulent flow (blue, left), with time rescaled to unity.

Figure 5

Temporal evolution of velocity components $v_x$ and $v_y$. Under laminar flow $v_y$ will remain close to zero. However, under turbulent flow, such as in this case, we observe motion that is orthogonal and even against the desired direction of motion (along the $+x$ direction).

Figure 6

Plot of crowd “pressure” overlaid with mean flow velocities for a time period of 3 s for (a) stop-and-go flow and (b) turbulent flow. Note the formation of irregular clusters as denoted by isocontour lines. Color bars show the range of “pressures” observed; note that values greater than $0.02$ do not arise in the occurrence of stop-and-go waves.

Figure 7

Scene setup for Love Parade simulations. The scene consists of three merging sets of agents, emerging from the inlets shown in the blue checkerboard pattern. Agents from the bottom two inlets proceed towards the third inlet at the top and vice versa, following the paths highlighted by the dotted arrows. Corridor dimensions are shown by solid red arrows.

Figure 8

(a) Plot of average speed vs density. (b) Plot of crowd “pressure.” We observe values greater than $0.02$ at local densities of $5.5$ people per ${\mathrm{m}}^2$ and higher, which are indicative of crowd turbulence.

Figure 9

Plot of crowd “pressure” overlaid with mean flow velocities for a time period of 3 s for turbulent flow. Note formation of irregular clusters as denoted by isocontour lines. Color bars show range of “pressures” observed.

Figure 10

Plot of crowd “pressure” $P\left(t\right)={\langle \rho \left(t\right){Var}_t\left(\mathrm{v}\right)\rangle }_{\mathrm{x}}$ in the Hajj scenario with both discomfort and friction (top), and with discomfort alone (bottom). ${\langle ·\rangle }_{\mathrm{x}}$ denotes the mean computed over the entire scene. Note how discomfort alone is unable to recreate turbulent flow conditions.

Figure 11

Plot of the magnitude of the strain rate tensor ${\parallel \nabla v+\left(\nabla v\right)}^T{\parallel }_F$ at $t=3$ s for turbulence flow in the Hajj scenario. Note cluster formation as shown by isocontours.